The Hanna Neumann conjecture for graphs of free groups with cyclic edge groups

The Hanna Neumann conjecture (HNC) for a free group G predicts that χ̄(U∩ V) χ̄ (U)χ̄(V) for all finitely generated subgroups U and V , where χ̄(H) = max\-χ(H),0\ denotes the reduced Euler characteristic of H . A strengthened version of the HNC was proved independently by Friedman and Mineyev in 2011. Recently, Antolín and Jaikin-Zapirain introduced the L² -Hall property and showed that if G is a hyperbolic limit group that satisfies this property, then G satisfies the HNC. Antolín and Jaikin-Zapirain established the L² -Hall property for free and surface groups, which Brown and Kharlampovich extended to all limit groups. In this paper, we prove the L² -Hall property for graphs of free groups with cyclic edge groups that are hyperbolic relative to virtually abelian subgroups. We also give another proof of the L² -Hall property for limit groups. As a corollary, we show that all these groups satisfy a strengthened version of the HNC.